LMFDB - Embedded newform 2160.3.l.g.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(161,2160)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2160.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	2160.161
Dual form			2160.3.l.g.161.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{5} +9.70820 q^{7} -14.1803i q^{11} -5.70820 q^{13} -5.94427i q^{17} -32.4164 q^{19} -1.58359i q^{23} -5.00000 q^{25} +2.06888i q^{29} -49.2492 q^{31} +21.7082i q^{35} -26.5836 q^{37} +65.2361i q^{41} -49.3738 q^{43} -70.1378i q^{47} +45.2492 q^{49} +28.6393i q^{53} +31.7082 q^{55} +95.2361i q^{59} +19.0000 q^{61} -12.7639i q^{65} -107.666 q^{67} +75.2624i q^{71} -96.7902 q^{73} -137.666i q^{77} +101.833 q^{79} +44.6656i q^{83} +13.2918 q^{85} -56.2918i q^{89} -55.4164 q^{91} -72.4853i q^{95} +132.584 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7} + 4 q^{13} - 76 q^{19} - 20 q^{25} - 36 q^{31} - 160 q^{37} + 44 q^{43} + 20 q^{49} + 100 q^{55} + 76 q^{61} - 216 q^{67} - 92 q^{73} + 300 q^{79} + 80 q^{85} - 168 q^{91} + 584 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 + 4 * q^13 - 76 * q^19 - 20 * q^25 - 36 * q^31 - 160 * q^37 + 44 * q^43 + 20 * q^49 + 100 * q^55 + 76 * q^61 - 216 * q^67 - 92 * q^73 + 300 * q^79 + 80 * q^85 - 168 * q^91 + 584 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	9.70820	1.38689	0.693443	−	0.720511i	$-0.256093\pi$
$7$	9.70820	1.38689	0.693443	+	0.720511i	$0.256093\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 14.1803i	− 1.28912i	−0.764553	−	0.644561i	$-0.777040\pi$
$11$	− 14.1803i	− 1.28912i	0.764553	−	0.644561i	$-0.222960\pi$
$12$	0	0
$13$	−5.70820	−0.439093	−0.219546	−	0.975602i	$-0.570458\pi$
$13$	−5.70820	−0.439093	−0.219546	+	0.975602i	$0.570458\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.94427i	− 0.349663i	−0.984598	−	0.174832i	$-0.944062\pi$
$17$	− 5.94427i	− 0.349663i	0.984598	−	0.174832i	$-0.0559381\pi$
$18$	0	0
$19$	−32.4164	−1.70613	−0.853063	−	0.521807i	$-0.825258\pi$
$19$	−32.4164	−1.70613	−0.853063	+	0.521807i	$0.825258\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.58359i	− 0.0688518i	−0.999407	−	0.0344259i	$-0.989040\pi$
$23$	− 1.58359i	− 0.0688518i	0.999407	−	0.0344259i	$-0.0109603\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.06888i	0.0713408i	0.999364	+	0.0356704i	$0.0113567\pi$
$29$	2.06888i	0.0713408i	−0.999364	+	0.0356704i	$0.988643\pi$
$30$	0	0
$31$	−49.2492	−1.58868	−0.794342	−	0.607470i	$-0.792184\pi$
$31$	−49.2492	−1.58868	−0.794342	+	0.607470i	$0.792184\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	21.7082i	0.620234i
$36$	0	0
$37$	−26.5836	−0.718475	−0.359238	−	0.933246i	$-0.616963\pi$
$37$	−26.5836	−0.718475	−0.359238	+	0.933246i	$0.616963\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	65.2361i	1.59112i	0.605872	+	0.795562i	$0.292824\pi$
$41$	65.2361i	1.59112i	−0.605872	+	0.795562i	$0.707176\pi$
$42$	0	0
$43$	−49.3738	−1.14823	−0.574114	−	0.818775i	$-0.694654\pi$
$43$	−49.3738	−1.14823	−0.574114	+	0.818775i	$0.694654\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 70.1378i	− 1.49229i	−0.665782	−	0.746146i	$-0.731902\pi$
$47$	− 70.1378i	− 1.49229i	0.665782	−	0.746146i	$-0.268098\pi$
$48$	0	0
$49$	45.2492	0.923454
$50$	0	0
$51$	0	0
$52$	0	0
$53$	28.6393i	0.540365i	0.962809	+	0.270182i	$0.0870840\pi$
$53$	28.6393i	0.540365i	−0.962809	+	0.270182i	$0.912916\pi$
$54$	0	0
$55$	31.7082	0.576513
$56$	0	0
$57$	0	0
$58$	0	0
$59$	95.2361i	1.61417i	0.590435	+	0.807085i	$0.298956\pi$
$59$	95.2361i	1.61417i	−0.590435	+	0.807085i	$0.701044\pi$
$60$	0	0
$61$	19.0000	0.311475	0.155738	−	0.987798i	$-0.450225\pi$
$61$	19.0000	0.311475	0.155738	+	0.987798i	$0.450225\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 12.7639i	− 0.196368i
$66$	0	0
$67$	−107.666	−1.60695	−0.803475	−	0.595339i	$-0.797018\pi$
$67$	−107.666	−1.60695	−0.803475	+	0.595339i	$0.797018\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	75.2624i	1.06003i	0.847987	+	0.530017i	$0.177814\pi$
$71$	75.2624i	1.06003i	−0.847987	+	0.530017i	$0.822186\pi$
$72$	0	0
$73$	−96.7902	−1.32589	−0.662947	−	0.748666i	$-0.730694\pi$
$73$	−96.7902	−1.32589	−0.662947	+	0.748666i	$0.730694\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 137.666i	− 1.78787i
$78$	0	0
$79$	101.833	1.28902	0.644511	−	0.764595i	$-0.277061\pi$
$79$	101.833	1.28902	0.644511	+	0.764595i	$0.277061\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	44.6656i	0.538140i	0.963121	+	0.269070i	$0.0867163\pi$
$83$	44.6656i	0.538140i	−0.963121	+	0.269070i	$0.913284\pi$
$84$	0	0
$85$	13.2918	0.156374
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 56.2918i	− 0.632492i	−0.948677	−	0.316246i	$-0.897577\pi$
$89$	− 56.2918i	− 0.632492i	0.948677	−	0.316246i	$-0.102423\pi$
$90$	0	0
$91$	−55.4164	−0.608972
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 72.4853i	− 0.763003i
$96$	0	0
$97$	132.584	1.36684	0.683421	−	0.730025i	$-0.260492\pi$
$97$	132.584	1.36684	0.683421	+	0.730025i	$0.260492\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	24.0000i	0.237624i	0.992917	+	0.118812i	$0.0379085\pi$
$101$	24.0000i	0.237624i	−0.992917	+	0.118812i	$0.962091\pi$
$102$	0	0
$103$	78.8328	0.765367	0.382684	−	0.923879i	$-0.375000\pi$
$103$	78.8328	0.765367	0.382684	+	0.923879i	$0.375000\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	59.6656i	0.557623i	0.960346	+	0.278811i	$0.0899404\pi$
$107$	59.6656i	0.557623i	−0.960346	+	0.278811i	$0.910060\pi$
$108$	0	0
$109$	−22.9149	−0.210228	−0.105114	−	0.994460i	$-0.533521\pi$
$109$	−22.9149	−0.210228	−0.105114	+	0.994460i	$0.533521\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	128.387i	1.13617i	0.822971	+	0.568084i	$0.192315\pi$
$113$	128.387i	1.13617i	−0.822971	+	0.568084i	$0.807685\pi$
$114$	0	0
$115$	3.54102	0.0307915
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 57.7082i	− 0.484943i
$120$	0	0
$121$	−80.0820	−0.661835
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	−40.7477	−0.320848	−0.160424	−	0.987048i	$-0.551286\pi$
$127$	−40.7477	−0.320848	−0.160424	+	0.987048i	$0.551286\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 260.164i	− 1.98599i	−0.118176	−	0.992993i	$-0.537705\pi$
$131$	− 260.164i	− 1.98599i	0.118176	−	0.992993i	$-0.462295\pi$
$132$	0	0
$133$	−314.705	−2.36620
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 164.331i	− 1.19950i	−0.800188	−	0.599749i	$-0.795267\pi$
$137$	− 164.331i	− 1.19950i	0.800188	−	0.599749i	$-0.204733\pi$
$138$	0	0
$139$	−34.4984	−0.248190	−0.124095	−	0.992270i	$-0.539603\pi$
$139$	−34.4984	−0.248190	−0.124095	+	0.992270i	$0.539603\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	80.9443i	0.566044i
$144$	0	0
$145$	−4.62616	−0.0319046
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 38.1803i	− 0.256244i	−0.991758	−	0.128122i	$-0.959105\pi$
$149$	− 38.1803i	− 0.256244i	0.991758	−	0.128122i	$-0.0408949\pi$
$150$	0	0
$151$	−182.997	−1.21190	−0.605950	−	0.795503i	$-0.707207\pi$
$151$	−182.997	−1.21190	−0.605950	+	0.795503i	$0.707207\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 110.125i	− 0.710481i
$156$	0	0
$157$	−144.705	−0.921689	−0.460844	−	0.887481i	$-0.652453\pi$
$157$	−144.705	−0.921689	−0.460844	+	0.887481i	$0.652453\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 15.3738i	− 0.0954897i
$162$	0	0
$163$	−16.8328	−0.103269	−0.0516344	−	0.998666i	$-0.516443\pi$
$163$	−16.8328	−0.103269	−0.0516344	+	0.998666i	$0.516443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 36.5805i	− 0.219045i	−0.993984	−	0.109522i	$-0.965068\pi$
$167$	− 36.5805i	− 0.219045i	0.993984	−	0.109522i	$-0.0349321\pi$
$168$	0	0
$169$	−136.416	−0.807198
$170$	0	0
$171$	0	0
$172$	0	0
$173$	32.3313i	0.186886i	0.995625	+	0.0934430i	$0.0297873\pi$
$173$	32.3313i	0.186886i	−0.995625	+	0.0934430i	$0.970213\pi$
$174$	0	0
$175$	−48.5410	−0.277377
$176$	0	0
$177$	0	0
$178$	0	0
$179$	136.774i	0.764100i	0.924142	+	0.382050i	$0.124782\pi$
$179$	136.774i	0.764100i	−0.924142	+	0.382050i	$0.875218\pi$
$180$	0	0
$181$	45.9180	0.253690	0.126845	−	0.991923i	$-0.459515\pi$
$181$	45.9180	0.253690	0.126845	+	0.991923i	$0.459515\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 59.4427i	− 0.321312i
$186$	0	0
$187$	−84.2918	−0.450758
$188$	0	0
$189$	0	0
$190$	0	0
$191$	68.3870i	0.358047i	0.983845	+	0.179024i	$0.0572938\pi$
$191$	68.3870i	0.358047i	−0.983845	+	0.179024i	$0.942706\pi$
$192$	0	0
$193$	77.7082	0.402633	0.201317	−	0.979526i	$-0.435478\pi$
$193$	77.7082	0.402633	0.201317	+	0.979526i	$0.435478\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	103.997i	0.527903i	0.964536	+	0.263952i	$0.0850259\pi$
$197$	103.997i	0.527903i	−0.964536	+	0.263952i	$0.914974\pi$
$198$	0	0
$199$	−239.751	−1.20478	−0.602389	−	0.798203i	$-0.705784\pi$
$199$	−239.751	−1.20478	−0.602389	+	0.798203i	$0.705784\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.0851i	0.0989416i
$204$	0	0
$205$	−145.872	−0.711572
$206$	0	0
$207$	0	0
$208$	0	0
$209$	459.676i	2.19941i
$210$	0	0
$211$	10.1672	0.0481857	0.0240929	−	0.999710i	$-0.492330\pi$
$211$	10.1672	0.0481857	0.0240929	+	0.999710i	$0.492330\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 110.403i	− 0.513503i
$216$	0	0
$217$	−478.122	−2.20332
$218$	0	0
$219$	0	0
$220$	0	0
$221$	33.9311i	0.153534i
$222$	0	0
$223$	−72.2492	−0.323988	−0.161994	−	0.986792i	$-0.551792\pi$
$223$	−72.2492	−0.323988	−0.161994	+	0.986792i	$0.551792\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.4752i	0.116631i	0.998298	+	0.0583155i	$0.0185729\pi$
$227$	26.4752i	0.116631i	−0.998298	+	0.0583155i	$0.981427\pi$
$228$	0	0
$229$	−280.495	−1.22487	−0.612435	−	0.790521i	$-0.709810\pi$
$229$	−280.495	−1.22487	−0.612435	+	0.790521i	$0.709810\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 115.082i	− 0.493914i	−0.969026	−	0.246957i	$-0.920569\pi$
$233$	− 115.082i	− 0.493914i	0.969026	−	0.246957i	$-0.0794307\pi$
$234$	0	0
$235$	156.833	0.667374
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 147.390i	− 0.616695i	−0.951274	−	0.308347i	$-0.900224\pi$
$239$	− 147.390i	− 0.616695i	0.951274	−	0.308347i	$-0.0997759\pi$
$240$	0	0
$241$	−62.5805	−0.259670	−0.129835	−	0.991536i	$-0.541445\pi$
$241$	−62.5805	−0.259670	−0.129835	+	0.991536i	$0.541445\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	101.180i	0.412981i
$246$	0	0
$247$	185.039	0.749148
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 309.580i	− 1.23339i	−0.787203	−	0.616694i	$-0.788472\pi$
$251$	− 309.580i	− 1.23339i	0.787203	−	0.616694i	$-0.211528\pi$
$252$	0	0
$253$	−22.4559	−0.0887584
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 75.7477i	− 0.294738i	−0.989082	−	0.147369i	$-0.952920\pi$
$257$	− 75.7477i	− 0.294738i	0.989082	−	0.147369i	$-0.0470805\pi$
$258$	0	0
$259$	−258.079	−0.996444
$260$	0	0
$261$	0	0
$262$	0	0
$263$	367.974i	1.39914i	0.714564	+	0.699570i	$0.246625\pi$
$263$	367.974i	1.39914i	−0.714564	+	0.699570i	$0.753375\pi$
$264$	0	0
$265$	−64.0395	−0.241658
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 279.358i	− 1.03850i	−0.854621	−	0.519252i	$-0.826211\pi$
$269$	− 279.358i	− 1.03850i	0.854621	−	0.519252i	$-0.173789\pi$
$270$	0	0
$271$	−62.4164	−0.230319	−0.115159	−	0.993347i	$-0.536738\pi$
$271$	−62.4164	−0.230319	−0.115159	+	0.993347i	$0.536738\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	70.9017i	0.257824i
$276$	0	0
$277$	26.6262	0.0961233	0.0480617	−	0.998844i	$-0.484696\pi$
$277$	26.6262	0.0961233	0.0480617	+	0.998844i	$0.484696\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	412.869i	1.46929i	0.678454	+	0.734643i	$0.262650\pi$
$281$	412.869i	1.46929i	−0.678454	+	0.734643i	$0.737350\pi$
$282$	0	0
$283$	−104.292	−0.368522	−0.184261	−	0.982877i	$-0.558989\pi$
$283$	−104.292	−0.368522	−0.184261	+	0.982877i	$0.558989\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	633.325i	2.20671i
$288$	0	0
$289$	253.666	0.877736
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 311.243i	− 1.06226i	−0.847289	−	0.531131i	$-0.821767\pi$
$293$	− 311.243i	− 1.06226i	0.847289	−	0.531131i	$-0.178233\pi$
$294$	0	0
$295$	−212.954	−0.721879
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.03947i	0.0302323i
$300$	0	0
$301$	−479.331	−1.59246
$302$	0	0
$303$	0	0
$304$	0	0
$305$	42.4853i	0.139296i
$306$	0	0
$307$	−454.158	−1.47934	−0.739671	−	0.672969i	$-0.765019\pi$
$307$	−454.158	−1.47934	−0.739671	+	0.672969i	$0.765019\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	211.957i	0.681535i	0.940148	+	0.340768i	$0.110687\pi$
$311$	211.957i	0.681535i	−0.940148	+	0.340768i	$0.889313\pi$
$312$	0	0
$313$	284.827	0.909989	0.454995	−	0.890494i	$-0.349641\pi$
$313$	284.827	0.909989	0.454995	+	0.890494i	$0.349641\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 252.056i	− 0.795128i	−0.917574	−	0.397564i	$-0.869856\pi$
$317$	− 252.056i	− 0.795128i	0.917574	−	0.397564i	$-0.130144\pi$
$318$	0	0
$319$	29.3375	0.0919670
$320$	0	0
$321$	0	0
$322$	0	0
$323$	192.692i	0.596569i
$324$	0	0
$325$	28.5410	0.0878185
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 680.912i	− 2.06964i
$330$	0	0
$331$	177.915	0.537507	0.268754	−	0.963209i	$-0.413388\pi$
$331$	177.915	0.537507	0.268754	+	0.963209i	$0.413388\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 240.748i	− 0.718650i
$336$	0	0
$337$	462.450	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$337$	462.450	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	698.371i	2.04801i
$342$	0	0
$343$	−36.4133	−0.106161
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 493.384i	− 1.42186i	−0.703265	−	0.710928i	$-0.748275\pi$
$347$	− 493.384i	− 1.42186i	0.703265	−	0.710928i	$-0.251725\pi$
$348$	0	0
$349$	62.0820	0.177885	0.0889427	−	0.996037i	$-0.471651\pi$
$349$	62.0820	0.177885	0.0889427	+	0.996037i	$0.471651\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	123.836i	0.350810i	0.984496	+	0.175405i	$0.0561235\pi$
$353$	123.836i	0.350810i	−0.984496	+	0.175405i	$0.943877\pi$
$354$	0	0
$355$	−168.292	−0.474061
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 243.994i	− 0.679648i	−0.940489	−	0.339824i	$-0.889632\pi$
$359$	− 243.994i	− 0.679648i	0.940489	−	0.339824i	$-0.110368\pi$
$360$	0	0
$361$	689.823	1.91087
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 216.430i	− 0.592958i
$366$	0	0
$367$	236.371	0.644062	0.322031	−	0.946729i	$-0.395634\pi$
$367$	236.371	0.644062	0.322031	+	0.946729i	$0.395634\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	278.036i	0.749424i
$372$	0	0
$373$	301.167	0.807419	0.403709	−	0.914887i	$-0.367721\pi$
$373$	301.167	0.807419	0.403709	+	0.914887i	$0.367721\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 11.8096i	− 0.0313252i
$378$	0	0
$379$	102.003	0.269137	0.134569	−	0.990904i	$-0.457035\pi$
$379$	102.003	0.269137	0.134569	+	0.990904i	$0.457035\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 705.158i	− 1.84114i	−0.390574	−	0.920572i	$-0.627723\pi$
$383$	− 705.158i	− 1.84114i	0.390574	−	0.920572i	$-0.372277\pi$
$384$	0	0
$385$	307.830	0.799558
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 346.043i	− 0.889570i	−0.895637	−	0.444785i	$-0.853280\pi$
$389$	− 346.043i	− 0.889570i	0.895637	−	0.444785i	$-0.146720\pi$
$390$	0	0
$391$	−9.41330	−0.0240749
$392$	0	0
$393$	0	0
$394$	0	0
$395$	227.705i	0.576469i
$396$	0	0
$397$	172.128	0.433571	0.216786	−	0.976219i	$-0.430443\pi$
$397$	172.128	0.433571	0.216786	+	0.976219i	$0.430443\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 436.407i	− 1.08830i	−0.838989	−	0.544148i	$-0.816853\pi$
$401$	− 436.407i	− 1.08830i	0.838989	−	0.544148i	$-0.183147\pi$
$402$	0	0
$403$	281.125	0.697580
$404$	0	0
$405$	0	0
$406$	0	0
$407$	376.964i	0.926202i
$408$	0	0
$409$	−353.663	−0.864701	−0.432350	−	0.901706i	$-0.642316\pi$
$409$	−353.663	−0.864701	−0.432350	+	0.901706i	$0.642316\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	924.571i	2.23867i
$414$	0	0
$415$	−99.8754	−0.240664
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 503.060i	− 1.20062i	−0.799768	−	0.600310i	$-0.795044\pi$
$419$	− 503.060i	− 1.20062i	0.799768	−	0.600310i	$-0.204956\pi$
$420$	0	0
$421$	−448.745	−1.06590	−0.532951	−	0.846146i	$-0.678917\pi$
$421$	−448.745	−1.06590	−0.532951	+	0.846146i	$0.678917\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	29.7214i	0.0699326i
$426$	0	0
$427$	184.456	0.431981
$428$	0	0
$429$	0	0
$430$	0	0
$431$	263.204i	0.610681i	0.952243	+	0.305341i	$0.0987703\pi$
$431$	263.204i	0.610681i	−0.952243	+	0.305341i	$0.901230\pi$
$432$	0	0
$433$	155.702	0.359589	0.179794	−	0.983704i	$-0.442457\pi$
$433$	155.702	0.359589	0.179794	+	0.983704i	$0.442457\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	51.3344i	0.117470i
$438$	0	0
$439$	−44.0031	−0.100235	−0.0501174	−	0.998743i	$-0.515960\pi$
$439$	−44.0031	−0.100235	−0.0501174	+	0.998743i	$0.515960\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	842.902i	1.90271i	0.308090	+	0.951357i	$0.400310\pi$
$443$	842.902i	1.90271i	−0.308090	+	0.951357i	$0.599690\pi$
$444$	0	0
$445$	125.872	0.282859
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 505.656i	− 1.12618i	−0.826395	−	0.563091i	$-0.809612\pi$
$449$	− 505.656i	− 1.12618i	0.826395	−	0.563091i	$-0.190388\pi$
$450$	0	0
$451$	925.070	2.05115
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 123.915i	− 0.272340i
$456$	0	0
$457$	414.577	0.907172	0.453586	−	0.891213i	$-0.350145\pi$
$457$	414.577	0.907172	0.453586	+	0.891213i	$0.350145\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 103.909i	− 0.225398i	−0.993629	−	0.112699i	$-0.964050\pi$
$461$	− 103.909i	− 0.225398i	0.993629	−	0.112699i	$-0.0359496\pi$
$462$	0	0
$463$	−814.997	−1.76025	−0.880126	−	0.474740i	$-0.842542\pi$
$463$	−814.997	−1.76025	−0.880126	+	0.474740i	$0.842542\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.8096i	0.0702561i	0.999383	+	0.0351281i	$0.0111839\pi$
$467$	32.8096i	0.0702561i	−0.999383	+	0.0351281i	$0.988816\pi$
$468$	0	0
$469$	−1045.24	−2.22866
$470$	0	0
$471$	0	0
$472$	0	0
$473$	700.138i	1.48021i
$474$	0	0
$475$	162.082	0.341225
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 574.456i	− 1.19928i	−0.800269	−	0.599641i	$-0.795310\pi$
$479$	− 574.456i	− 1.19928i	0.800269	−	0.599641i	$-0.204690\pi$
$480$	0	0
$481$	151.745	0.315477
$482$	0	0
$483$	0	0
$484$	0	0
$485$	296.466i	0.611270i
$486$	0	0
$487$	−189.131	−0.388359	−0.194179	−	0.980966i	$-0.562204\pi$
$487$	−189.131	−0.388359	−0.194179	+	0.980966i	$0.562204\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 453.437i	− 0.923496i	−0.887011	−	0.461748i	$-0.847222\pi$
$491$	− 453.437i	− 0.923496i	0.887011	−	0.461748i	$-0.152778\pi$
$492$	0	0
$493$	12.2980	0.0249452
$494$	0	0
$495$	0	0
$496$	0	0
$497$	730.663i	1.47015i
$498$	0	0
$499$	−100.088	−0.200578	−0.100289	−	0.994958i	$-0.531977\pi$
$499$	−100.088	−0.200578	−0.100289	+	0.994958i	$0.531977\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 564.692i	− 1.12265i	−0.827596	−	0.561324i	$-0.810292\pi$
$503$	− 564.692i	− 1.12265i	0.827596	−	0.561324i	$-0.189708\pi$
$504$	0	0
$505$	−53.6656	−0.106269
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 839.522i	− 1.64935i	−0.565603	−	0.824677i	$-0.691357\pi$
$509$	− 839.522i	− 1.64935i	0.565603	−	0.824677i	$-0.308643\pi$
$510$	0	0
$511$	−939.659	−1.83886
$512$	0	0
$513$	0	0
$514$	0	0
$515$	176.276i	0.342283i
$516$	0	0
$517$	−994.577	−1.92375
$518$	0	0
$519$	0	0
$520$	0	0
$521$	380.227i	0.729802i	0.931046	+	0.364901i	$0.118897\pi$
$521$	380.227i	0.729802i	−0.931046	+	0.364901i	$0.881103\pi$
$522$	0	0
$523$	194.207	0.371332	0.185666	−	0.982613i	$-0.440556\pi$
$523$	194.207	0.371332	0.185666	+	0.982613i	$0.440556\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	292.751i	0.555504i
$528$	0	0
$529$	526.492	0.995259
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 372.381i	− 0.698651i
$534$	0	0
$535$	−133.416	−0.249376
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 641.649i	− 1.19044i
$540$	0	0
$541$	159.836	0.295445	0.147723	−	0.989029i	$-0.452806\pi$
$541$	159.836	0.295445	0.147723	+	0.989029i	$0.452806\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 51.2392i	− 0.0940168i
$546$	0	0
$547$	647.295	1.18335	0.591677	−	0.806175i	$-0.298466\pi$
$547$	647.295	1.18335	0.591677	+	0.806175i	$0.298466\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 67.0658i	− 0.121716i
$552$	0	0
$553$	988.614	1.78773
$554$	0	0
$555$	0	0
$556$	0	0
$557$	901.161i	1.61788i	0.587889	+	0.808942i	$0.299959\pi$
$557$	901.161i	1.61788i	−0.587889	+	0.808942i	$0.700041\pi$
$558$	0	0
$559$	281.836	0.504179
$560$	0	0
$561$	0	0
$562$	0	0
$563$	529.003i	0.939615i	0.882769	+	0.469807i	$0.155677\pi$
$563$	529.003i	0.939615i	−0.882769	+	0.469807i	$0.844323\pi$
$564$	0	0
$565$	−287.082	−0.508110
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 493.623i	− 0.867527i	−0.901027	−	0.433764i	$-0.857185\pi$
$569$	− 493.623i	− 0.867527i	0.901027	−	0.433764i	$-0.142815\pi$
$570$	0	0
$571$	760.246	1.33143	0.665715	−	0.746206i	$-0.268127\pi$
$571$	760.246	1.33143	0.665715	+	0.746206i	$0.268127\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.91796i	0.0137704i
$576$	0	0
$577$	−38.4559	−0.0666480	−0.0333240	−	0.999445i	$-0.510609\pi$
$577$	−38.4559	−0.0666480	−0.0333240	+	0.999445i	$0.510609\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	433.623i	0.746339i
$582$	0	0
$583$	406.115	0.696596
$584$	0	0
$585$	0	0
$586$	0	0
$587$	548.489i	0.934394i	0.884153	+	0.467197i	$0.154736\pi$
$587$	548.489i	0.934394i	−0.884153	+	0.467197i	$0.845264\pi$
$588$	0	0
$589$	1596.48	2.71050
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 9.97058i	− 0.0168138i	−0.999965	−	0.00840690i	$-0.997324\pi$
$593$	− 9.97058i	− 0.0168138i	0.999965	−	0.00840690i	$-0.00267603\pi$
$594$	0	0
$595$	129.039	0.216873
$596$	0	0
$597$	0	0
$598$	0	0
$599$	507.755i	0.847670i	0.905739	+	0.423835i	$0.139316\pi$
$599$	507.755i	0.847670i	−0.905739	+	0.423835i	$0.860684\pi$
$600$	0	0
$601$	−680.416	−1.13214	−0.566070	−	0.824357i	$-0.691537\pi$
$601$	−680.416	−1.13214	−0.566070	+	0.824357i	$0.691537\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 179.069i	− 0.295982i
$606$	0	0
$607$	305.374	0.503087	0.251544	−	0.967846i	$-0.419062\pi$
$607$	305.374	0.503087	0.251544	+	0.967846i	$0.419062\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	400.361i	0.655255i
$612$	0	0
$613$	−876.584	−1.42999	−0.714995	−	0.699130i	$-0.753571\pi$
$613$	−876.584	−1.42999	−0.714995	+	0.699130i	$0.753571\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 740.489i	− 1.20014i	−0.799946	−	0.600072i	$-0.795138\pi$
$617$	− 740.489i	− 1.20014i	0.799946	−	0.600072i	$-0.204862\pi$
$618$	0	0
$619$	309.902	0.500650	0.250325	−	0.968162i	$-0.419463\pi$
$619$	309.902	0.500650	0.250325	+	0.968162i	$0.419463\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 546.492i	− 0.877195i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	158.020i	0.251224i
$630$	0	0
$631$	324.830	0.514786	0.257393	−	0.966307i	$-0.417137\pi$
$631$	324.830	0.514786	0.257393	+	0.966307i	$0.417137\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 91.1146i	− 0.143487i
$636$	0	0
$637$	−258.292	−0.405482
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 951.246i	− 1.48400i	−0.670398	−	0.742002i	$-0.733877\pi$
$641$	− 951.246i	− 1.48400i	0.670398	−	0.742002i	$-0.266123\pi$
$642$	0	0
$643$	−835.289	−1.29905	−0.649525	−	0.760341i	$-0.725032\pi$
$643$	−835.289	−1.29905	−0.649525	+	0.760341i	$0.725032\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 247.393i	− 0.382370i	−0.981554	−	0.191185i	$-0.938767\pi$
$647$	− 247.393i	− 0.382370i	0.981554	−	0.191185i	$-0.0612330\pi$
$648$	0	0
$649$	1350.48	2.08086
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 228.293i	− 0.349606i	−0.984603	−	0.174803i	$-0.944071\pi$
$653$	− 228.293i	− 0.349606i	0.984603	−	0.174803i	$-0.0559288\pi$
$654$	0	0
$655$	581.745	0.888160
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 907.512i	− 1.37710i	−0.725187	−	0.688552i	$-0.758247\pi$
$659$	− 907.512i	− 1.37710i	0.725187	−	0.688552i	$-0.241753\pi$
$660$	0	0
$661$	−468.152	−0.708248	−0.354124	−	0.935199i	$-0.615221\pi$
$661$	−468.152	−0.708248	−0.354124	+	0.935199i	$0.615221\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 703.702i	− 1.05820i
$666$	0	0
$667$	3.27627	0.00491195
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 269.426i	− 0.401530i
$672$	0	0
$673$	−195.629	−0.290682	−0.145341	−	0.989382i	$-0.546428\pi$
$673$	−195.629	−0.290682	−0.145341	+	0.989382i	$0.546428\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	114.158i	0.168623i	0.996439	+	0.0843116i	$0.0268691\pi$
$677$	114.158i	0.168623i	−0.996439	+	0.0843116i	$0.973131\pi$
$678$	0	0
$679$	1287.15	1.89565
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1240.50i	1.81625i	0.418705	+	0.908123i	$0.362484\pi$
$683$	1240.50i	1.81625i	−0.418705	+	0.908123i	$0.637516\pi$
$684$	0	0
$685$	367.456	0.536432
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 163.479i	− 0.237270i
$690$	0	0
$691$	−487.669	−0.705743	−0.352872	−	0.935672i	$-0.614795\pi$
$691$	−487.669	−0.705743	−0.352872	+	0.935672i	$0.614795\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 77.1409i	− 0.110994i
$696$	0	0
$697$	387.781	0.556357
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 925.384i	− 1.32009i	−0.751226	−	0.660046i	$-0.770537\pi$
$701$	− 925.384i	− 1.32009i	0.751226	−	0.660046i	$-0.229463\pi$
$702$	0	0
$703$	861.745	1.22581
$704$	0	0
$705$	0	0
$706$	0	0
$707$	232.997i	0.329557i
$708$	0	0
$709$	−616.322	−0.869283	−0.434642	−	0.900603i	$-0.643125\pi$
$709$	−616.322	−0.869283	−0.434642	+	0.900603i	$0.643125\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	77.9907i	0.109384i
$714$	0	0
$715$	−180.997	−0.253143
$716$	0	0
$717$	0	0
$718$	0	0
$719$	404.067i	0.561984i	0.959710	+	0.280992i	$0.0906634\pi$
$719$	404.067i	0.561984i	−0.959710	+	0.280992i	$0.909337\pi$
$720$	0	0
$721$	765.325	1.06148
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 10.3444i	− 0.0142682i
$726$	0	0
$727$	834.991	1.14854	0.574271	−	0.818665i	$-0.305285\pi$
$727$	834.991	1.14854	0.574271	+	0.818665i	$0.305285\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	293.492i	0.401493i
$732$	0	0
$733$	−678.480	−0.925620	−0.462810	−	0.886457i	$-0.653159\pi$
$733$	−678.480	−0.925620	−0.462810	+	0.886457i	$0.653159\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1526.74i	2.07155i
$738$	0	0
$739$	284.580	0.385089	0.192544	−	0.981288i	$-0.438326\pi$
$739$	284.580	0.385089	0.192544	+	0.981288i	$0.438326\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	489.947i	0.659418i	0.944083	+	0.329709i	$0.106951\pi$
$743$	489.947i	0.659418i	−0.944083	+	0.329709i	$0.893049\pi$
$744$	0	0
$745$	85.3738	0.114596
$746$	0	0
$747$	0	0
$748$	0	0
$749$	579.246i	0.773359i
$750$	0	0
$751$	1180.49	1.57189	0.785945	−	0.618297i	$-0.212177\pi$
$751$	1180.49	1.57189	0.785945	+	0.618297i	$0.212177\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 409.193i	− 0.541978i
$756$	0	0
$757$	358.255	0.473257	0.236628	−	0.971600i	$-0.423958\pi$
$757$	358.255	0.473257	0.236628	+	0.971600i	$0.423958\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 428.115i	− 0.562569i	−0.959624	−	0.281285i	$-0.909239\pi$
$761$	− 428.115i	− 0.562569i	0.959624	−	0.281285i	$-0.0907605\pi$
$762$	0	0
$763$	−222.462	−0.291562
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 543.627i	− 0.708770i
$768$	0	0
$769$	421.656	0.548318	0.274159	−	0.961684i	$-0.411601\pi$
$769$	421.656	0.548318	0.274159	+	0.961684i	$0.411601\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	792.246i	1.02490i	0.858718	+	0.512449i	$0.171262\pi$
$773$	792.246i	1.02490i	−0.858718	+	0.512449i	$0.828738\pi$
$774$	0	0
$775$	246.246	0.317737
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2114.72i	− 2.71466i
$780$	0	0
$781$	1067.25	1.36651
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 323.570i	− 0.412192i
$786$	0	0
$787$	−218.456	−0.277581	−0.138790	−	0.990322i	$-0.544321\pi$
$787$	−218.456	−0.277581	−0.138790	+	0.990322i	$0.544321\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1246.41i	1.57574i
$792$	0	0
$793$	−108.456	−0.136767
$794$	0	0
$795$	0	0
$796$	0	0
$797$	78.5341i	0.0985371i	0.998786	+	0.0492686i	$0.0156890\pi$
$797$	78.5341i	0.0985371i	−0.998786	+	0.0492686i	$0.984311\pi$
$798$	0	0
$799$	−416.918	−0.521800
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1372.52i	1.70924i
$804$	0	0
$805$	34.3769	0.0427043
$806$	0	0
$807$	0	0
$808$	0	0
$809$	922.233i	1.13997i	0.821656	+	0.569983i	$0.193050\pi$
$809$	922.233i	1.13997i	−0.821656	+	0.569983i	$0.806950\pi$
$810$	0	0
$811$	−1199.82	−1.47944	−0.739719	−	0.672916i	$-0.765041\pi$
$811$	−1199.82	−1.47944	−0.739719	+	0.672916i	$0.765041\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 37.6393i	− 0.0461832i
$816$	0	0
$817$	1600.52	1.95902
$818$	0	0
$819$	0	0
$820$	0	0
$821$	460.663i	0.561099i	0.959840	+	0.280550i	$0.0905167\pi$
$821$	460.663i	0.561099i	−0.959840	+	0.280550i	$0.909483\pi$
$822$	0	0
$823$	1375.20	1.67096	0.835478	−	0.549523i	$-0.185191\pi$
$823$	1375.20	1.67096	0.835478	+	0.549523i	$0.185191\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 497.800i	− 0.601935i	−0.953634	−	0.300968i	$-0.902690\pi$
$827$	− 497.800i	− 0.601935i	0.953634	−	0.300968i	$-0.0973096\pi$
$828$	0	0
$829$	−1339.24	−1.61549	−0.807744	−	0.589533i	$-0.799312\pi$
$829$	−1339.24	−1.61549	−0.807744	+	0.589533i	$0.799312\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 268.974i	− 0.322898i
$834$	0	0
$835$	81.7965	0.0979598
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 297.613i	− 0.354723i	−0.984146	−	0.177362i	$-0.943244\pi$
$839$	− 297.613i	− 0.354723i	0.984146	−	0.177362i	$-0.0567563\pi$
$840$	0	0
$841$	836.720	0.994910
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 305.036i	− 0.360990i
$846$	0	0
$847$	−777.453	−0.917890
$848$	0	0
$849$	0	0
$850$	0	0
$851$	42.0976i	0.0494684i
$852$	0	0
$853$	887.647	1.04062	0.520309	−	0.853978i	$-0.325817\pi$
$853$	887.647	1.04062	0.520309	+	0.853978i	$0.325817\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	623.977i	0.728094i	0.931381	+	0.364047i	$0.118605\pi$
$857$	623.977i	0.728094i	−0.931381	+	0.364047i	$0.881395\pi$
$858$	0	0
$859$	1015.24	1.18189	0.590945	−	0.806712i	$-0.298755\pi$
$859$	1015.24	1.18189	0.590945	+	0.806712i	$0.298755\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1480.72i	1.71578i	0.513833	+	0.857890i	$0.328225\pi$
$863$	1480.72i	1.71578i	−0.513833	+	0.857890i	$0.671775\pi$
$864$	0	0
$865$	−72.2949	−0.0835779
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1444.02i	− 1.66171i
$870$	0	0
$871$	614.577	0.705600
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 108.541i	− 0.124047i
$876$	0	0
$877$	−253.957	−0.289575	−0.144788	−	0.989463i	$-0.546250\pi$
$877$	−253.957	−0.289575	−0.144788	+	0.989463i	$0.546250\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	490.502i	0.556756i	0.960472	+	0.278378i	$0.0897968\pi$
$881$	490.502i	0.556756i	−0.960472	+	0.278378i	$0.910203\pi$
$882$	0	0
$883$	881.878	0.998730	0.499365	−	0.866392i	$-0.333567\pi$
$883$	881.878	0.998730	0.499365	+	0.866392i	$0.333567\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 267.827i	− 0.301947i	−0.988538	−	0.150973i	$-0.951759\pi$
$887$	− 267.827i	− 0.301947i	0.988538	−	0.150973i	$-0.0482407\pi$
$888$	0	0
$889$	−395.587	−0.444979
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2273.61i	2.54604i
$894$	0	0
$895$	−305.836	−0.341716
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 101.891i	− 0.113338i
$900$	0	0
$901$	170.240	0.188946
$902$	0	0
$903$	0	0
$904$	0	0
$905$	102.676i	0.113454i
$906$	0	0
$907$	−493.240	−0.543815	−0.271907	−	0.962323i	$-0.587654\pi$
$907$	−493.240	−0.543815	−0.271907	+	0.962323i	$0.587654\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1798.73i	1.97445i	0.159328	+	0.987226i	$0.449067\pi$
$911$	1798.73i	1.97445i	−0.159328	+	0.987226i	$0.550933\pi$
$912$	0	0
$913$	633.374	0.693728
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2525.73i	− 2.75434i
$918$	0	0
$919$	338.000	0.367791	0.183896	−	0.982946i	$-0.441129\pi$
$919$	338.000	0.367791	0.183896	+	0.982946i	$0.441129\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 429.613i	− 0.465453i
$924$	0	0
$925$	132.918	0.143695
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1205.17i	− 1.29728i	−0.761096	−	0.648639i	$-0.775339\pi$
$929$	− 1205.17i	− 1.29728i	0.761096	−	0.648639i	$-0.224661\pi$
$930$	0	0
$931$	−1466.82	−1.57553
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 188.482i	− 0.201585i
$936$	0	0
$937$	1187.65	1.26750	0.633750	−	0.773538i	$-0.281515\pi$
$937$	1187.65	1.26750	0.633750	+	0.773538i	$0.281515\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1037.42i	1.10247i	0.834350	+	0.551235i	$0.185843\pi$
$941$	1037.42i	1.10247i	−0.834350	+	0.551235i	$0.814157\pi$
$942$	0	0
$943$	103.307	0.109552
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 304.180i	− 0.321203i	−0.987019	−	0.160602i	$-0.948657\pi$
$947$	− 304.180i	− 0.321203i	0.987019	−	0.160602i	$-0.0513435\pi$
$948$	0	0
$949$	552.498	0.582190
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1308.97i	1.37353i	0.726881	+	0.686763i	$0.240969\pi$
$953$	1308.97i	1.37353i	−0.726881	+	0.686763i	$0.759031\pi$
$954$	0	0
$955$	−152.918	−0.160124
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1595.36i	− 1.66357i
$960$	0	0
$961$	1464.49	1.52392
$962$	0	0
$963$	0	0
$964$	0	0
$965$	173.761i	0.180063i
$966$	0	0
$967$	−769.489	−0.795749	−0.397874	−	0.917440i	$-0.630252\pi$
$967$	−769.489	−0.795749	−0.397874	+	0.917440i	$0.630252\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	73.4164i	0.0756091i	0.999285	+	0.0378045i	$0.0120364\pi$
$971$	73.4164i	0.0756091i	−0.999285	+	0.0378045i	$0.987964\pi$
$972$	0	0
$973$	−334.918	−0.344212
$974$	0	0
$975$	0	0
$976$	0	0
$977$	865.384i	0.885756i	0.896582	+	0.442878i	$0.146043\pi$
$977$	865.384i	0.885756i	−0.896582	+	0.442878i	$0.853957\pi$
$978$	0	0
$979$	−798.237	−0.815359
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1728.98i	− 1.75888i	−0.476013	−	0.879438i	$-0.657919\pi$
$983$	− 1728.98i	− 1.75888i	0.476013	−	0.879438i	$-0.342081\pi$
$984$	0	0
$985$	−232.544	−0.236085
$986$	0	0
$987$	0	0
$988$	0	0
$989$	78.1880i	0.0790577i
$990$	0	0
$991$	−445.675	−0.449722	−0.224861	−	0.974391i	$-0.572193\pi$
$991$	−445.675	−0.449722	−0.224861	+	0.974391i	$0.572193\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 536.099i	− 0.538793i
$996$	0	0
$997$	78.2253	0.0784607	0.0392303	−	0.999230i	$-0.487509\pi$
$997$	78.2253	0.0784607	0.0392303	+	0.999230i	$0.487509\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.l.g.161.4		4
3.2	odd	2	inner	2160.3.l.g.161.2		4
4.3	odd	2		135.3.c.c.26.4	yes	4
12.11	even	2		135.3.c.c.26.1	✓	4
20.3	even	4		675.3.d.i.674.4		4
20.7	even	4		675.3.d.e.674.1		4
20.19	odd	2		675.3.c.p.26.1		4
36.7	odd	6		405.3.i.c.296.4		8
36.11	even	6		405.3.i.c.296.1		8
36.23	even	6		405.3.i.c.26.4		8
36.31	odd	6		405.3.i.c.26.1		8
60.23	odd	4		675.3.d.e.674.2		4
60.47	odd	4		675.3.d.i.674.3		4
60.59	even	2		675.3.c.p.26.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.3.c.c.26.1	✓	4	12.11	even	2
135.3.c.c.26.4	yes	4	4.3	odd	2
405.3.i.c.26.1		8	36.31	odd	6
405.3.i.c.26.4		8	36.23	even	6
405.3.i.c.296.1		8	36.11	even	6
405.3.i.c.296.4		8	36.7	odd	6
675.3.c.p.26.1		4	20.19	odd	2
675.3.c.p.26.4		4	60.59	even	2
675.3.d.e.674.1		4	20.7	even	4
675.3.d.e.674.2		4	60.23	odd	4
675.3.d.i.674.3		4	60.47	odd	4
675.3.d.i.674.4		4	20.3	even	4
2160.3.l.g.161.2		4	3.2	odd	2	inner
2160.3.l.g.161.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{5} +9.70820 q^{7} -14.1803i q^{11} -5.70820 q^{13} -5.94427i q^{17} -32.4164 q^{19} -1.58359i q^{23} -5.00000 q^{25} +2.06888i q^{29} -49.2492 q^{31} +21.7082i q^{35} -26.5836 q^{37} +65.2361i q^{41} -49.3738 q^{43} -70.1378i q^{47} +45.2492 q^{49} +28.6393i q^{53} +31.7082 q^{55} +95.2361i q^{59} +19.0000 q^{61} -12.7639i q^{65} -107.666 q^{67} +75.2624i q^{71} -96.7902 q^{73} -137.666i q^{77} +101.833 q^{79} +44.6656i q^{83} +13.2918 q^{85} -56.2918i q^{89} -55.4164 q^{91} -72.4853i q^{95} +132.584 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7} + 4 q^{13} - 76 q^{19} - 20 q^{25} - 36 q^{31} - 160 q^{37} + 44 q^{43} + 20 q^{49} + 100 q^{55} + 76 q^{61} - 216 q^{67} - 92 q^{73} + 300 q^{79} + 80 q^{85} - 168 q^{91} + 584 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 + 4 * q^13 - 76 * q^19 - 20 * q^25 - 36 * q^31 - 160 * q^37 + 44 * q^43 + 20 * q^49 + 100 * q^55 + 76 * q^61 - 216 * q^67 - 92 * q^73 + 300 * q^79 + 80 * q^85 - 168 * q^91 + 584 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more